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The geometrically nonlinear dynamic responses of simply supported beams under moving loads
Accepted Manuscript
The geometrically nonlinear dynamic responses of simply supported beams under moving loads G.G. Sheng , X. Wang PII: DOI: Reference:
S0307-904X(17)30259-7 10.1016/j.apm.2017.03.064 APM 11705
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
10 October 2016 17 January 2017 28 March 2017
Please cite this article as: G.G. Sheng , X. Wang , The geometrically nonlinear dynamic responses of simply supported beams under moving loads, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.064
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Highlights The load-structure interaction model with consideration of the nonlinear effect
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is created.
The effects of key parameters on the nonlinear vibration are investigated.
A series of comparison are performed and the investigations demonstrate good reliability
The dynamic design of beams will be improved by the nonlinear analysis
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and results.
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The geometrically nonlinear dynamic responses of simply supported beams under moving loads G.G. Sheng
, X.Wang
b
School of Civil Engineering and Architecture, Changsha University of Science and
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a
a,*
Technology, Changsha, Hunan 410114, People’s Republic of China b
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China
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Abstract
This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as a
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Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load-structure system is discussed, and Kelvin-Voigt material model
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is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's
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principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by
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using the Newmark method and Newton-Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass
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and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability. Keywords：Nonlinear vibration; beam; moving load 1. Introduction Dynamic analysis of beams under moving loads has been an important topic in structural engineering. The dynamic responses (inclinding dynamic stresses) could *
Corresponding author. E-mail: [email protected] 2
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become very larger than those of the static loads. The theoretical principles of dynamic interaction are well known [1]. Numerous works on vibration of beams under moving loads are reported in the literature [2, 3]. Linear free and forced vibrations of beams under moving loads have been studied extensively. Using Euler-Bernoulli beam hypothesis, Law and Zhu [4] developed the dynamic responses of a continuous beam under a moving vehicle by considering the
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interaction between the beam and the vehicle. The vehicle is modeled as a four freedom degrees mass-spring system. Based on the Biot's consolidation theory and Timoshenko beam model, Keivan et al.[5] analysed the dynamic response of poroelastic beams acted upon by a moving point load accounting for shear
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deformation. Using the modal superposition, Yang and Lin [6] studied the dynamic interaction between the vehicle and the beam, and the responses of the beam and the vehicle are obtained. Museros et al.[7] studied the vibrations of simply supported beams under a constant moving loads, and a new approximate approach for estimating
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the maximum acceleration was proposed. Using the mode superposition, Sudheesh Kumar et al.[8] proposed a simple and compact formula to determine the free
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vibration responses of a uniform beam under a single moving load. Using Laplace transformation, Johansson et al. [9] obtained a closed-form solution for the vibration
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of Bernoulli-Euler beams subjected to a constant moving loads. Based on Timoshenko beam theory, Ding et al. [10] investigated the dynamic responses of a Timoshenko
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beam under a moving harmonic load. Zhu et al. [11] established a linear complementarity method for a vehicle-bridge dynamic system considering separation
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and random roughness. The dynamic interaction between the vehicle and the bridge is transformed into a standard linear complementarity problem. Using Euler-Bernoulli beam hypothesis, Dimitrovová [12] obtained a new formula for the critical velocity of a uniformly moving load. It is assumed that the load is traversing a beam supported by a foundation of a finite depth. Simplified plane models of the foundation are presented for the finite and infinite beams, respectively. When beams are subjected to large magnitude loads or the frequency of loads is close to the natural frequency of the beam, the beam may vibrate at large amplitude. 3
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In this case, linear theory is not suitable to analyze the large amplitude vibration of beams. To design a stable and reliable beam, we should analyze the nonlinear vibration characteristics of beams. Based on the Euler-Bernoulli beam theory and the von Kármán nonlinear strain-displacement relationship, He et al. [13] developed the nonlinear vibration of carbon nanotubes (CNTs)/fiber/polymer laminated multiscale composite beam, Şimşek [14] investigated the nonlinear vibration of microbeams with
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the nonlinear elastic foundation, and Zhu and Chung [15] studied the nonlinear dynamical behaviors of a deploying beam with a spinning motion. Based on the Euler-Bernoulli beam model and Hamilton’s principle, Peng et al. [16] analysed the size-dependent micro-beams with nonlinear elasticity under electrical actuation.
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Based on the Timoshenko beam theory, Ritz method and von Kármán type nonlinear strain-displacement relationships, Chen et al. [17] analysed the nonlinear vibration of shear deformable sandwich beam with a functionally graded porous core. Using the von-Kármán nonlinear strain-displacement relations, Faraji Oskouie and Ansari [18]
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explored the nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams based on the Gurtin–Murdoch surface stress theory and the Galerkin approach. Based
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on the Rayleigh beam theory with von Kármán type nonlinear strain-displacement relationships, Domagalski and Jędrysiak [19] analysed the nonlinear vibrations of
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beams with periodic structure.
Although beams under moving loads have been widely investigated, only limited
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literature can be found about the nonlinear vibration of beams subjected to moving loads in the research works. An experimental study [20] was carried out on a
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T-section reinforced concrete beam subject to the action of a moving model vehicle, and nonlinearities were detected by examining the changes in the instantaneous frequency when the vehicular loads are at different locations along the beam. Based on the von Kármán nonlinear theory, Şimşek and Kocatürk [21] investigated the nonlinear vibration of a damping beam under a moving load. Using the separation of variables approach and Euler-Bernoulli beam hypothesis, Mamandi et al. [22] investigated the nonlinear problem of an inclined beam under a moving force. Based on Euler-Bernoulli beam hypothesis and von Kármán geometric nonlinear theory, Tao 4
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et al. [23] studied the nonlinear dynamic behaviors of fiber metal laminated beams under moving loads in thermal environments. Using the large deformation assumption in beam theory, Karimi and Ziaei-Rad [24] explored the nonlinear coupled vibration of a beam with moving supports under the action of a moving mass, and two different cases (constant and variable speed) were studied. From the above-mentioned literatures of the nonlinear problem, it is found that
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most of moving loads are constant (moving mass, moving force model). The objective of this paper is to analyze the nonlinear dynamic interaction problem of the load-structure system. The displacement field of the beam is expressed in terms of the linear fundamental vibration modes. Galerkin’s method is utilized to convert the
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governing partial differential equations to nonlinear ordinary differential equations. The effects of prestress load, internal damping, surface roughness, span of the beam and moving speed of the load on the nonlinear vibration responses are investigated.
2.1. Loading model
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2. Theoretical formulations
A load is considered as a vehicle (four freedom degrees system moving at a
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constant speed V , see Fig. 1). m1 , m2 and m3 is the mass of the vehicle body
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and the vehicle axles, respectively. I1 is the inertia moment of the vehicle body. ( k1 , k3 ) and ( k 2 , k 4 ) are the stiffnesses from the suspensions and the tyres, ( c1 , c3 ) and ( c2 , c4 ) are the damping from the suspensions and the tyres,
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respectively.
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respectively. u1 and are the vertical displacement and pitch angle of the vehicle body, respectively. u2 and u3 are the vertical displacements of the vehicle axles. The structure is modeled as a Euler-Benoulli beam with rectanglar cross section and simply supported ends, length L , and subjected to axial prestress load P . Using the D'Alembert's principle, the equations of motion of loading model are derived as follows:
m1u1 k1 z2 c1 z2 k3 z3 c3 z3 m1 g 0
(1) 5
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I1 k3e2 z3 c3e2 z3 k1e1 z2 c1e1 z2 0
(2)
m2u2 k3 z2 c1 z2 f1 m2 g 0
(3)
m3u3 k1 z3 c3 z3 f 2 m3 g 0
(4)
where the superposed
dot
denotes differentiation with
respect
to
time,
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z 2 u 2 (u1 e1 ) , z3 u3 (u1 e2 ) , g gravitational acceleration, f1 and f 2 are the load-structure interaction forces, and
f1 k4 (w1 r1 u2 ) c2 (w1 r1 u2 )
(5)
f 2 k2 (w2 r2 u3 ) c4 (w2 r2 u3 )
(6)
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w1 and w2 are the vertical dynamic deflections of the beam at the contact points of the rear wheel and the front wheel, respectively. r1 and r2 are the surface roughness of the beam at the contact points of the rear tyre and the front tyre, respectively.
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2.2. Beam model
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Let us consider a differential element of the beam shown in Fig.2. f ( x, t ) is transverse load along the length of the beam (interaction force at wheel). FN is the
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axial force, M the bending moment, which are known as the stress resultants, and
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they are defined in terms of the normal stress x on a cross section as
FN x dA
(7)
M x zdA
(8)
A
A
When rotary inertia and shear deformation are omitted, the shear force is
Considering
Fs the
pure
M x bending
(9) case,
the
von
Kármán
type
nonlinear
strain-displacement relation is given as
1 w 2 2w ) z 2 2 x x
x (
(10)
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where w is the transverse displacement of the beam. The material of the beam can be expressed in terms of the Kelvin-Voigt model, so the normal stress x is given as
x E ( x
x ) t
(11)
where E is Young’s modulus, the internal damping constant of the beam.
M EI
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Using Eqs. (7)–(11), the stress resultants( M , Fs , FN ) can be written as
2w 3w EI x 2 x 2 t
(12)
3w 4w Fs EI 3 EI 3 x x t
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(13)
1 w w 2 w F N EA ( ) 2 EA( )P 2 x x xt
(14)
where A , I are the area and inertia moment of the beam cross section, respectively.
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Using D'Alembert's principle, we derive the equilibrium equation of the beam. Summing the forces on an element of the beam ( in the z – direction, see Fig. 2)
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gives the following equilibrium equation:
2 w Fs ( FN ) f ( x, t ) 0 t 2 x x
(15)
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A
w x
(16)
f ( x, t ) f1 ( x x1 ) H1 (t ) f 2 ( x x2 ) H 2 (t )
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where is the mass of the beam per unit volume, and
(17)
L where x1 Vt , x2 Vt , H1 (t ) H t 1 H t , e1 e2 (wheelbase),
V1
V
L H 2 (t ) H (t ) 1 H t , (.) denotes Dirac delta function, H (t ) denotes V
Heaviside unit step function. Substituting Eqs. (13) ,(14), (16) and (17) into Eq. (15), the nonlinear equation of motion can be written as 7
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A
2w 4w 5 w 2w EI EI P t 2 x 4 x 4t x 2 3 w 2 2 w w 2 w 2 w w 2 3 w EA( ) 2 2 EA( 2 ) EA[( ) 2 ] f ( x, t ) 2 x x x x xt x x t
(18)
where f ( x, t ) f1 ( x x1 ) H1 (t ) f 2 ( x x2 ) H 2 (t ) . Using Eqs. (1) - (4), f1 and f 2
f1 m2 g
m1 ge2
m2u2
m1u1e2
I1
f 2 m3 g
m1 ge1
m3u3
m1u1e1
I1
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can be rewritten as (19)
(20)
2.3. Discretization equations of the load-structure interaction
Using the separation of variables approach, the nonlinear responses of the beam are given by the expression [22, 25]: n
w( x, t ) i ( x)qi (t )
(21)
M
i 1
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where qi (t ) are time-dependent generalized coordinates to be determined, i (x) are the linear fundamental vibration modes, and n is the number of vibration modes.
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For the present case, the normal modal functions are
2 i sin x , i 1,2,3n AL L
(22)
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i ( x)
The vertical displacements, w1 and w2 [see Eqs (5) and (6)], at the contact points
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of the wheels and the beam are not independent coordinates, which can be given below
w1 w( x, t ) x x , 1
w1 V
w2 w( x, t ) x x , w2 V 2
w( x, t ) w( x, t ) , x x x t x x 1 1
w( x, t ) w( x, t ) x x x t x x 2 2
(23)
(24)
Substituting Eqs.(19)–(22) into Eq. (18) , Eqs. (5),(6),(23),(24) into Eqs. (1)–(4),
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and using Galerkin procedure, the coupled nonlinear ordinary differential equations are obtained in terms of generalized coordinates qi (t ) ( i 1,2,3n ), u1 , , u2 and u3 :
qi (t ) 2ii qi (t ) i2 qi (t )
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[e2i ( x1 ) H1 (t ) e1i ( x2 ) H 2 (t ) ] m1u1 i ( x1 ) H1 (t )m2u2
i ( x2 ) H 2 (t )m3u3 [i ( x1 ) H1 (t ) i ( x2 ) H 2 (t ) ]I1 n
n
n
n
n
n
nol nol k ijkl q j (t )q k (t )ql (t ) cijkl q j (t )q k (t )q l (t ) j 1 k 1 l 1
j 1 k 1 l 1
2 i sin (Vt ) AL L
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(m2 g m1 ge2 ) H1 (t )
(m3 g m1 ge1 ) H 2 (t )
2 i sin Vt AL L
(25)
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i 1,2,3n
m1u1 (c1 c3 )u1 (c1e1 c3e2 ) c1u2 c3u3
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(k1 k3 )u1 (k1e1 k3e2 ) k1u2 k3u3 m1g
(26)
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I1 (c1e1 c3e2 )u1 (c1e12 c3e22 ) c1e1u2 c3e2u3
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(k1e1 k3e2 )u1 (k1e12 k3e22 ) k1e1u2 k3e2u3 0
(27)
n
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m2u2 c2 i ( x1 )qi (t ) c1u1 c1e1 (c1 c2 )u2 k1u1 k1e1 i 1
n k2i ( x1 ) c2V i( x1 )qi (t ) (k1 k2 )u2 m2 g k2 r1 c2r1 i 1
(28)
n
m3u3 c4 i ( x2 )qi (t ) c3u1 c3e2 (c3 c4 )u3 k3u1 k3e2 i 1
n k4i ( x2 ) c4V i( x2 )qi (t ) (k3 k4 )u3 m3 g k4 r2 c4r2 i 1
(29)
where
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i i L
2 EI P L , 1 A EI i
2
i EI i , 2i L A 4
and the nonlinear term coefficients are given by
nol cijkl
3E L i ( x) j ( x) k ( x) l( x)dx 2 0 2 E
L
0
i ( x) j ( x)k( x)l( x)dx
E L ( x) j ( x)k ( x)l( x)dx 0 i
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nol k ijkl
Making the appropriate substitutions and performing the integrations, the coupled nonlinear equations of motion can be written as
MX + (CL + CNol )X + (K L + K Nol )X = F
(30)
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L where M , C and K L are the linear time-dependent mass, damping and stiffness
matrices (i. e., time varying model), respectively, X is the vector of generalized coordinates, given by
qn u1 u2 u3
T
(31)
M
X q1 q2
CNol and K Nol contain the nonlinear terms, which are the nonlinear damping and
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nonlinear stiffness matrix, respectively, and F is the generalised force matrix. Eq. (30) consists of n 4 second-order nonlinear ordinary differential equations,
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which are solved in the time domain by the Newmark method and the Newton–Raphson iteration method [21]. For the linear time varying model, the
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equations of motion can also be obtained from Eq. (30) by setting the nonlinear terms
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to zero.
2.4. Surface roughness Surface roughness is one of the important factors of dynamic excitation. The
surface roughness can be represented with a random process that can be described by a power spectral density function. In the present study, the power spectral density function for road surface roughness is given as [26, 27]:
S () S (0 )(
2 ) 0
(32)
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where the spatial frequency (cycle/m), S () the power spectral density, 0 -
the reference spatial frequency ( 0 0.1 m 1), S (0 ) is related to the pavement quality, its value is prescribed in Table 1. Surface profiles, r ( x) , are generated as the sum of a series of harmonics: N
r ( x) ak cos(2k x k )
where ak 2S (k ) ,
(33)
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k 1
max min , K min k , min the N -
minimum spatial frequency ( min 0.01m 1), max the maximum spatial frequency ( max 10 m 1), k is the random phase angle uniformly distributed in the interval
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-
[0, 2 ] .
According to Eq. (33), the surface roughness of the beam at the contact points of
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the rear tyre and the front tyre can be obtained as:
r1 (t ) r ( x) x x , r1 (t ) V
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1
r2 (t ) r ( x) x x , r2 (t ) V
(34)
dr ( x) dx x x 2
(35)
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2
dr ( x) dx x x 1
Using Eqs. (34) and (35), the generalized forces caused by the surface roughness can
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be determined in Eqs. (28) and (29). 3. Numerical simulations and results
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3.1 Simple validation of the present method
Example 1 In this section, we present some results about a mass traversing a proportionally damped beam (see Fig.3). The beam is subject to two forces: the weight and the inertia force of moving mass m . From Eqs.(18)-(20), the linear governing equation of the moving mass model is easy to obtain by neglecting the nonlinear terms
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2w w EI 4 w mg m d 2 w( x, t ) C [ ] ( x Vt ) H 2 (t ) d t 2 t A x 4 A A dt 2 where C d
(36)
D , D is the damping coefficient. The linear time varying model can A
be solved by using Galerkin procedure and the Newmark method. This model is similar to the linear model developed by Pesterev and Bergman [28]. The numerical
EI 275.4408m 4 / s 2 , A
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values of the system parameters were as follows: L 6 m,
m 0.2 . Three variants of damping level were considered: (a) C d 0. , (b) AL
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C d 2. , and (c) C d 9. The modes number n used in all of the computations is 3. The displacements curves ( w( , t ) , Vt ) corresponding to these cases are depicted in Fig.4. The comparison shows excellent agreement with Pesterev and Bergman [28].
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Example 2 The moving oscillator model shown in Fig.5 is considered in this section. Using Eqs. (1)–(4), the equations of motion of the oscillator model are derived as
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follows:
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m1z1 c( z1 z 2 ) k ( z1 z 2 ) m1 g f ( x, t )
(38)
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m2 z2 c( z1 z 2 ) k ( z1 z 2 ) m2 g
(37)
where m1 and m 2 are non-suspended and suspended mass of the oscillator model,
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respectively, c the damping coefficient related to suspension, k the stiffness coefficient related to suspension, f ( x, t ) ( x Vt ) the interaction force at the wheel, and z1 and z 2 is the vertical displacement of the non-suspended mass m1 and suspended mass m 2 , respectively. From Eq.(18), the linear governing equation of the beam is easy to obtain by neglecting the nonlinear terms
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A
2w w 4w C EI f ( x, t ) ( x Vt ) H 2 (t ) d t 2 t x 4
(39)
Using Eqs. (37) and (38), the interaction force is given as
f ( x, t ) (m1 m2 ) g (m1 z1 m2 z2 )
(40)
where the vertical displacement, z1 , at the contact point of the oscillator model and
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the beam is not independent coordinate, which can be given below
z1 w( x, t ) x Vt w( x, t ) x
z1 V 2
x Vt
2 w( x, t ) x 2
w( x, t ) t
2V
x Vt
(42)
x Vt
2 w( x, t ) xt
2 w( x, t ) t 2
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z1 V
(41)
x Vt
(43)
x Vt
Numerical results, based on the coupled linear time varying mode (see Eq.(30), setting the nonlinear terms to zero), have been computed and plotted in order to
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establish reasonable comparisons. The following dimensionless parameters [11] are used:
(m1 m2 ) ) , 0.3 ( b1 , v v AL
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(
EI V , b1 2 ) , b1 L AL4
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0.5 (
0 0.25 ( 0
m1 ) , 0.5 m2
k1 c ) , v 0.125 ( v ) , m2 2m 2 v
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and the modal damping ratio is set to be 0.02. The response magnification factor is
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used in the results, which is defined as
Dd
w( x, t ) Dst
( Dst
(m1 m2 ) gL3 ) 48EI
(44)
As can be seen from the comparisons (see Fig.6), the agreement with those in the literature [11] is good. 3.2 Nonlinear vibration of the load-structure interaction Several examples of applications of the methodology described in the above are presented here. A sensitivity study was conducted to analyze the effect of key
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parameters on the nonlinear vibration of the system. These parameters included the roughness coefficient, the span, prestress, internal damping of the beam and moving speed of the load. Table 2 was the mechanical properties used in all of the computations unless otherwise specified. Pcr
2 EI L2
is Euler’s buckling load. In all
of numerical examples, zero initial conditions were assumed.
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The linear and nonlinear dynamic responses computed for the beam and vehicle have been plotted in Figs. 7 and 8. Fig. 7 shows the effect of the nonlinear on the displacements, velocities and accelerations of the beam at the midspan. Fig. 8 shows the effect of the nonlinear on the vertical displacements, vertical velocities and
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vertical accelerations of the vehicle body. It can be seen that the dynamic responses of the nonlinear analysis is higher than the one obtained from the linear solution. This phenomenon was also noted elsewhere [22]. This incident is known as the softening behavior which is mostly due to the existence of the cubic non-linearities in the
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equations of motion of the load-structure system, and hence this system is taken to be equivalent to a nonlinear soft spring [25]. A free vibration with damping (decay) can
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also be observed after the load departs from the beam in Fig. 7. Form this, one may also conclude that the nonlinear analyses are more expensive than the linear approach
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with respect to both storage and CPU time. For different surface roughness of the beam, Fig. 9 displays the nonlinear
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acceleration responses of the beam at the midspan and the vertical acceleration of the vehicle body. It is observed that the nonlinear acceleration responses of the beam and
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the vertical acceleration of the vehicle body increase as the surface roughness increases (road class:B→C→D, see Table 1). This is due to the fact that the dynamic excitation increases as the surface roughness coefficient increases. The impact factor is an important parameter in the design of beams, and can be defined [30] as follows:
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Rd R s Rs
(45)
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where Rd and Rs are the absolute maximum responses at the midspan from the dynamic and static studies respectively. Fig.10 shows that the impact factor at the midspan decreases as the span L of the beam increases. The impact factor decreases rapidly as the span L increases from 0 to 14m. However, when L is larger than 16m, the impact factor changes very slowly, and the impact factors are almost same for
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different internal damping of the beam. From this figure, it can also be said that by increasing the value of internal damping , the impact factor at the midspan decreases which is generally a natural phenomenon in any structural system. Designers may obtain the desirable dynamic characteristics adequate to design
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purpose as they choose the span L and internal damping appropriately.
Fig.11 illustrates the linear and nonlinear dynamic responses at the midspan of the beam for different prestress loads. It is clear to see that the impact factor at the midspan increases with the increase of the prestress load. This phenomenon was also noted
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elsewhere [31]. When P / Pcr >0.35, the difference between the linear and the
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nonlinear model increases with the increase of the prestress loads. This is due to the fact that the stiffness of the beam will be reduced due to the influence of prestress, so
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the beam is softened and the nonlinearity of the system increases. The impact factor is sensitive to the moving speed of the load. Fig.12 shows the
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effect of the moving speed of the load on the impact factor. In this figure, the load velocity ranges from 5m/s to 85m/s. The maximum impact factor shows a peak,
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which correspond to the critical velocity Vcr 50 m/s. It can be said that in the under critical velocity ( V < Vcr ) the impact factor of the beam generally increases by increasing the moving speed of the load, and in the overcritical velocity ( V > Vcr ), the impact factor decreases by increasing the moving speed of the load. The phenomenon is also reported in linear moving force model [7]. According to the governing equations (25), the critical velocity can also be defined as
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Vcr
L ( i 1,2,3 ) i
(46)
where is the natural frequency of the load-structure system (see Eqs.(25)-(29), neglecting the nonlinear terms ). The result calculated from Eq. (46) is agreement with the numerical result in Fig.12. 4. Conclusions
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This paper focuses on the nonlinear vibration and dynamic interaction of the beam under moving loads. The numerical results show that the dynamic responses of the nonlinear analysis are higher than the one obtained from the linear solution, and the nonlinear responses of the beam and the vehicle body increase as the surface
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roughness increases. The results also confirm that the impact factors decrease rapidly as the span and the damping increase. However, when the span is larger, the impact factor changes very slowly, and the impact factors are almost same for larger damping. Another finding is that the nonlinear dynamic responses, and the difference between the the linear and the nonlinear model increase with the increasing of the prestress
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load. Moreover, the critical velocity can be observed by the simulation results. The
of beams.
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Acknowledgements
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meaningful results in the present paper are helpful for the application and the design
The authors thank the supports of the Hunan Provincial Natural Science Foundation
CE
of China under No. 13JJ4053.
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simply-supported beam bridges under moving loads: Maximum resonance, cancellation and resonant vertical acceleration, J. Sound Vib. 332 (2013) 326 –345.
[8] C.P. Sudheesh Kumar, C. Sujatha, K. Shankar, Vibration of simply supported
M
beams under a single moving load: A detailed study of cancellation phenomenon, Int. J. Mech. Sci. 99 (2015) 40–47.
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[9] C. Johansson, C. Pacoste, R. Karoumi, Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by
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concentrated moving loads, Comput. Struct. 119 (2013) 85–94. [10] H. Ding, K.l. Shi, L.Q. Chen, S.P. Yang, Adomian polynomials for
nonlinear
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response of supported Timoshenko beams subjected to a moving harmonic load, Acta Mech Solida Sin 27 (2014) 383–393.
AC
[11] D.Y. Zhu, Y.H. Zhang, H.A. Ouyang, Linear complementarity method for dynamic analysis of bridges under moving vehicles considering separation and surface roughness, Comput. Struct. 154 (2015) 135 –144.
[12] Z. Dimitrovová, Critical velocity of a uniformly moving load on a beam supported by a finite depth foundation, J. Sound. Vib. 366 (2016) 325–342. [13] X.Q. He, M. Rafiee, S. Mareishi, K.M. Liew, Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams, Compos. Struct. 131 (2015) 1111–1123. 17
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[14] M. Şimşek,
Nonlinear static and free vibration analysis of microbeams based
on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Compos. Struct. 112 (2014) 264–272. [15] Kefei Zhu, Jintai Chung, Nonlinear lateral vibrations of a deploying Euler–Bernoulli beam with a spinning motion, Int. J. Mech. Sci. 90 (2015) 200–212.
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[16] Jianshe Peng, Liu Yang, Fan Lin, Jie Yang, Dynamic analysis of size-dependent micro-beams with nonlinear elasticity under electrical actuation, Appl. Math. Model. 43 (2017) 441–453.
[17] Da Chen, Sritawat Kitipornchai, Jie Yang, Nonlinear free vibration of shear
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deformable sandwich beam with a functionally graded porous core, Thin-Walled Struct. 107 (2016) 39–48.
[18] M. Faraji Oskouie, R. Ansari, Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects, Appl.
M
Math. Model. 43 (2017) 337–350.
[19] Łukasz Domagalski, Jarosław Jędrysiak, Geometrically nonlinear vibrations of
ED
slender meso-periodic beams.The tolerance modeling approach, Compos. Struct. 136 (2016) 270–277.
PT
[20] S.S. Law, X.Q. Zhu, Nonlinear Characteristics of Damaged Concrete Structures under Vehicular Load, J. Struct. Eng. 131(8) (2005) 1277–1285.
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[21] M. Şimşek, T. Kocatürk, Nonlinear dynamic analysis of an eccentrically prestressed damped beam under a concentrated moving harmonic load, J. Sound
AC
Vib. 320 (2009) 235–253.
[22] A. Mamandi, M.H. Kargarnovin, D. Younesian, Nonlinear dynamics of an inclined beam subjected to a moving load, Nonlinear Dyn. 60 (2010) 277–293.
[23] C. Tao, Y.M. Fu, H.L. Dai, Nonlinear dynamic analysis of fiber metal laminated beams subjected to moving loads in thermal environment, Compos. Struct. 140 (2016) 410 – 416. [24] A. H. Karimi, S. Ziaei-Rad, Vibration analysis of a beam with moving support subjected to a moving mass traveling with constant and variable speed, Commun. 18
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Nonlinear Sci. Numer. Simulat. 29 (2015) 372–390. [25] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, 1979. [26] O. Javier, M.G. José, A. Pablo, Á.A. Miguel, Relevance of a complete road surface description in vehicle-bridge interaction dynamics, Eng. Struct. 56 (2013) 466–476.
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[27] ISO-8608. Mechanical vibration-road surface profiles-reporting of measured data.
[28] A.V. Pesterev, L.A. Bergman, Response of a nonconservative continuous system to a moving co ncentrated load, J. Appl. Mech. 65 (1998) 436 – 444.
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[29] S.S. Law, J.Q. Bu, X.Q. Zhu, S.L. Chan, Vehicle axle loads identification using finite element method, Eng. Struct. 26 (2004) 1143–1153.
[30] X.Q. Zhu, S.S. Law, Dynamic load on continuous multi-lane bridge deck from moving vehicles, J. Sound Vib. 251 (2002) 697–716.
M
[31] H. Zhong, M.J. Yang, Z.L. Gao (Jerry), Dynamic responses of prestressed bridge and vehicle through bridge-vehicle interaction analysis, Eng. Struct. 87 (2015)
AC
CE
PT
ED
116–125.
19
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PT
ED
M
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Fig. 1. Model of the load-structure system.
AC
CE
Fig. 2 Differential element of the beam of length dx .
Fig. 3. Model of the moving mass system.
20
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0.30
(a )
w( , t) (m)
0.25
Present work Pesterev et al.( 1998)
0.20 0.15 0.10
0.25
(b )
0.2
w( , t) (m)
0.15 0.10
M
0.05 0.2
0.4
w( , t) (m)
AC
CE
0.12
0.6
0.8
1.0
0.6
0.8
1.0
0.8
1.0
t
ED PT 0.15
(c )
t
Present work Pesterev et al.( 1998)
0.20
0.00 0.0
0.4
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0.00 0.0
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0.05
Present work Pesterev et al.( 1998)
0.09 0.06 0.03 0.00 0.0
0.2
0.4
0.6 t
Fig. 4 Displacements of the mass moving with constant speed v 6 m/s on the simply supported beam: (a) C d 0.
(b) C d 2.
(c) C d 9. 21
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1.8 1.5
Present work Zhu et al. (2015)
1.2 0.9 0.6 0.3
PT
ED
0.0 0.0
M
Response factor Dd
(a )
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Fig.5. Model of the moving oscillator system.
0.2
0.4
0.6
0.8
1.0
0.8
1.0
t
2.5
AC
CE
Response factor Dd
(b )
2.0
Present work Zhu et al. (2015)
1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6 t
Fig. 6. Response magnification factors of system: (a) displacement at the contact point of the wheel and the beam, (b) displacement at the midspan of the beam.
22
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0.04
(a )
-0.04 linear nonlinear
-0.08 -0.12 0.0
0.5
1.0
1.5 t (s)
2.5
3.0
PT
ED
M
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(b )
2.0
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w( L/2,t) (m)
0.00
AC
CE
(c)
Fig. 7. Time history responses of the beam midspan: (a) vertical displacement, (b) vertical velocity, (c) vertical acceleration.
23
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(a )
linear nonlinear
0.00
u1(m)
-0.05 -0.10
-0.20 0.0
0.3
0.6
0.9 t (s)
1.2
1.5
PT
ED
M
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(b )
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-0.15
AC
CE
(c )
Fig. 8. Time history responses of the vehicle body: (a) vertical displacement, (b) vertical velocity, (c) vertical acceleration.
24
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(a )
CE
PT
ED
M
(b)
AC
Fig. 9 Effect of the surface roughness of the beam on the time history responses: (a) vertical acceleration of the vehicle body, (b) vertical acceleration of the beam midspan.
25
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= 0.05 = 0.07 = 0.09
IP
0.9
0.6
0.3 12
15 L (m)
18
21
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9
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1.2
Fig.10. Effects of the span L and internal damping of the beam on impact
ED
M
factors.
PT
0.6
0.4
AC
IP
CE
0.5
linear nonlinear
0.3 0.2 0.1
0.2
0.3
0.4
P/Pcr
Fig.11. Effect of the prestress load of the beam on impact factors.
26
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1.0 linear
nonlinear
0.8
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IP
0.6 0.4 0.2
10 20 30 40 50 60 70 80 90 V (m/s)
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0
AC
CE
PT
ED
M
Fig.12. Effect of the moving speed of the load on impact factors.
27
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Table 1. Classification of road roughness [26]. S (0 ) (m3)
road class
16×10
B (Good)
64×10
-6 -6
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A (Very good)
C (Medium)
256×10
-6
1024×10
E (Very poor)
4096×10
-6 -6
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D (Poor)
Table 2. Parameters of the load-structure system [21, 29]. Beam
Vehicle
I1 1.47 105 kgm2
m1 17735 kg
4.27 m
e1 2.05 m
e2 2.22 m
m2 1000 kg
0.02 s
m3 1500 kg
k1 2.47 106 N/m
P 0.2 Pcr
k2 3.74 106 N/m
k3 4.23 106 N/m
A 0.36 m2
k4 4.60 106 N/m
c1 4.00 104 N/m/s
I 0.0192 m4
c2 4.30 103 N/m/s
c3 3.00 104 N/m/s
S (0 ) 64 106 m3
c4 3.90 103 N/m/s
V 15 m/s
AC
CE
PT
A 1500 kg/m
ED
E 35 Gpa
M
L 20 m
28
The geometrically nonlinear dynamic responses of simply supported beams under moving loads G.G. Sheng , X. Wang PII: DOI: Reference:
S0307-904X(17)30259-7 10.1016/j.apm.2017.03.064 APM 11705
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
10 October 2016 17 January 2017 28 March 2017
Please cite this article as: G.G. Sheng , X. Wang , The geometrically nonlinear dynamic responses of simply supported beams under moving loads, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.064
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Highlights The load-structure interaction model with consideration of the nonlinear effect
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is created.
The effects of key parameters on the nonlinear vibration are investigated.
A series of comparison are performed and the investigations demonstrate good reliability
The dynamic design of beams will be improved by the nonlinear analysis
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AC
CE
PT
ED
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and results.
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The geometrically nonlinear dynamic responses of simply supported beams under moving loads G.G. Sheng
, X.Wang
b
School of Civil Engineering and Architecture, Changsha University of Science and
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a
a,*
Technology, Changsha, Hunan 410114, People’s Republic of China b
School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China
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Abstract
This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as a
M
Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load-structure system is discussed, and Kelvin-Voigt material model
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is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's
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principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by
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using the Newmark method and Newton-Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass
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and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability. Keywords：Nonlinear vibration; beam; moving load 1. Introduction Dynamic analysis of beams under moving loads has been an important topic in structural engineering. The dynamic responses (inclinding dynamic stresses) could *
Corresponding author. E-mail: [email protected] 2
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become very larger than those of the static loads. The theoretical principles of dynamic interaction are well known [1]. Numerous works on vibration of beams under moving loads are reported in the literature [2, 3]. Linear free and forced vibrations of beams under moving loads have been studied extensively. Using Euler-Bernoulli beam hypothesis, Law and Zhu [4] developed the dynamic responses of a continuous beam under a moving vehicle by considering the
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interaction between the beam and the vehicle. The vehicle is modeled as a four freedom degrees mass-spring system. Based on the Biot's consolidation theory and Timoshenko beam model, Keivan et al.[5] analysed the dynamic response of poroelastic beams acted upon by a moving point load accounting for shear
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deformation. Using the modal superposition, Yang and Lin [6] studied the dynamic interaction between the vehicle and the beam, and the responses of the beam and the vehicle are obtained. Museros et al.[7] studied the vibrations of simply supported beams under a constant moving loads, and a new approximate approach for estimating
M
the maximum acceleration was proposed. Using the mode superposition, Sudheesh Kumar et al.[8] proposed a simple and compact formula to determine the free
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vibration responses of a uniform beam under a single moving load. Using Laplace transformation, Johansson et al. [9] obtained a closed-form solution for the vibration
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of Bernoulli-Euler beams subjected to a constant moving loads. Based on Timoshenko beam theory, Ding et al. [10] investigated the dynamic responses of a Timoshenko
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beam under a moving harmonic load. Zhu et al. [11] established a linear complementarity method for a vehicle-bridge dynamic system considering separation
AC
and random roughness. The dynamic interaction between the vehicle and the bridge is transformed into a standard linear complementarity problem. Using Euler-Bernoulli beam hypothesis, Dimitrovová [12] obtained a new formula for the critical velocity of a uniformly moving load. It is assumed that the load is traversing a beam supported by a foundation of a finite depth. Simplified plane models of the foundation are presented for the finite and infinite beams, respectively. When beams are subjected to large magnitude loads or the frequency of loads is close to the natural frequency of the beam, the beam may vibrate at large amplitude. 3
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In this case, linear theory is not suitable to analyze the large amplitude vibration of beams. To design a stable and reliable beam, we should analyze the nonlinear vibration characteristics of beams. Based on the Euler-Bernoulli beam theory and the von Kármán nonlinear strain-displacement relationship, He et al. [13] developed the nonlinear vibration of carbon nanotubes (CNTs)/fiber/polymer laminated multiscale composite beam, Şimşek [14] investigated the nonlinear vibration of microbeams with
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the nonlinear elastic foundation, and Zhu and Chung [15] studied the nonlinear dynamical behaviors of a deploying beam with a spinning motion. Based on the Euler-Bernoulli beam model and Hamilton’s principle, Peng et al. [16] analysed the size-dependent micro-beams with nonlinear elasticity under electrical actuation.
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Based on the Timoshenko beam theory, Ritz method and von Kármán type nonlinear strain-displacement relationships, Chen et al. [17] analysed the nonlinear vibration of shear deformable sandwich beam with a functionally graded porous core. Using the von-Kármán nonlinear strain-displacement relations, Faraji Oskouie and Ansari [18]
M
explored the nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams based on the Gurtin–Murdoch surface stress theory and the Galerkin approach. Based
ED
on the Rayleigh beam theory with von Kármán type nonlinear strain-displacement relationships, Domagalski and Jędrysiak [19] analysed the nonlinear vibrations of
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beams with periodic structure.
Although beams under moving loads have been widely investigated, only limited
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literature can be found about the nonlinear vibration of beams subjected to moving loads in the research works. An experimental study [20] was carried out on a
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T-section reinforced concrete beam subject to the action of a moving model vehicle, and nonlinearities were detected by examining the changes in the instantaneous frequency when the vehicular loads are at different locations along the beam. Based on the von Kármán nonlinear theory, Şimşek and Kocatürk [21] investigated the nonlinear vibration of a damping beam under a moving load. Using the separation of variables approach and Euler-Bernoulli beam hypothesis, Mamandi et al. [22] investigated the nonlinear problem of an inclined beam under a moving force. Based on Euler-Bernoulli beam hypothesis and von Kármán geometric nonlinear theory, Tao 4
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et al. [23] studied the nonlinear dynamic behaviors of fiber metal laminated beams under moving loads in thermal environments. Using the large deformation assumption in beam theory, Karimi and Ziaei-Rad [24] explored the nonlinear coupled vibration of a beam with moving supports under the action of a moving mass, and two different cases (constant and variable speed) were studied. From the above-mentioned literatures of the nonlinear problem, it is found that
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most of moving loads are constant (moving mass, moving force model). The objective of this paper is to analyze the nonlinear dynamic interaction problem of the load-structure system. The displacement field of the beam is expressed in terms of the linear fundamental vibration modes. Galerkin’s method is utilized to convert the
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governing partial differential equations to nonlinear ordinary differential equations. The effects of prestress load, internal damping, surface roughness, span of the beam and moving speed of the load on the nonlinear vibration responses are investigated.
2.1. Loading model
M
2. Theoretical formulations
A load is considered as a vehicle (four freedom degrees system moving at a
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constant speed V , see Fig. 1). m1 , m2 and m3 is the mass of the vehicle body
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and the vehicle axles, respectively. I1 is the inertia moment of the vehicle body. ( k1 , k3 ) and ( k 2 , k 4 ) are the stiffnesses from the suspensions and the tyres, ( c1 , c3 ) and ( c2 , c4 ) are the damping from the suspensions and the tyres,
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respectively.
AC
respectively. u1 and are the vertical displacement and pitch angle of the vehicle body, respectively. u2 and u3 are the vertical displacements of the vehicle axles. The structure is modeled as a Euler-Benoulli beam with rectanglar cross section and simply supported ends, length L , and subjected to axial prestress load P . Using the D'Alembert's principle, the equations of motion of loading model are derived as follows:
m1u1 k1 z2 c1 z2 k3 z3 c3 z3 m1 g 0
(1) 5
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I1 k3e2 z3 c3e2 z3 k1e1 z2 c1e1 z2 0
(2)
m2u2 k3 z2 c1 z2 f1 m2 g 0
(3)
m3u3 k1 z3 c3 z3 f 2 m3 g 0
(4)
where the superposed
dot
denotes differentiation with
respect
to
time,
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z 2 u 2 (u1 e1 ) , z3 u3 (u1 e2 ) , g gravitational acceleration, f1 and f 2 are the load-structure interaction forces, and
f1 k4 (w1 r1 u2 ) c2 (w1 r1 u2 )
(5)
f 2 k2 (w2 r2 u3 ) c4 (w2 r2 u3 )
(6)
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w1 and w2 are the vertical dynamic deflections of the beam at the contact points of the rear wheel and the front wheel, respectively. r1 and r2 are the surface roughness of the beam at the contact points of the rear tyre and the front tyre, respectively.
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2.2. Beam model
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Let us consider a differential element of the beam shown in Fig.2. f ( x, t ) is transverse load along the length of the beam (interaction force at wheel). FN is the
PT
axial force, M the bending moment, which are known as the stress resultants, and
AC
CE
they are defined in terms of the normal stress x on a cross section as
FN x dA
(7)
M x zdA
(8)
A
A
When rotary inertia and shear deformation are omitted, the shear force is
Considering
Fs the
pure
M x bending
(9) case,
the
von
Kármán
type
nonlinear
strain-displacement relation is given as
1 w 2 2w ) z 2 2 x x
x (
(10)
6
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where w is the transverse displacement of the beam. The material of the beam can be expressed in terms of the Kelvin-Voigt model, so the normal stress x is given as
x E ( x
x ) t
(11)
where E is Young’s modulus, the internal damping constant of the beam.
M EI
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Using Eqs. (7)–(11), the stress resultants( M , Fs , FN ) can be written as
2w 3w EI x 2 x 2 t
(12)
3w 4w Fs EI 3 EI 3 x x t
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(13)
1 w w 2 w F N EA ( ) 2 EA( )P 2 x x xt
(14)
where A , I are the area and inertia moment of the beam cross section, respectively.
M
Using D'Alembert's principle, we derive the equilibrium equation of the beam. Summing the forces on an element of the beam ( in the z – direction, see Fig. 2)
ED
gives the following equilibrium equation:
2 w Fs ( FN ) f ( x, t ) 0 t 2 x x
(15)
PT
A
w x
(16)
f ( x, t ) f1 ( x x1 ) H1 (t ) f 2 ( x x2 ) H 2 (t )
AC
CE
where is the mass of the beam per unit volume, and
(17)
L where x1 Vt , x2 Vt , H1 (t ) H t 1 H t , e1 e2 (wheelbase),
V1
V
L H 2 (t ) H (t ) 1 H t , (.) denotes Dirac delta function, H (t ) denotes V
Heaviside unit step function. Substituting Eqs. (13) ,(14), (16) and (17) into Eq. (15), the nonlinear equation of motion can be written as 7
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A
2w 4w 5 w 2w EI EI P t 2 x 4 x 4t x 2 3 w 2 2 w w 2 w 2 w w 2 3 w EA( ) 2 2 EA( 2 ) EA[( ) 2 ] f ( x, t ) 2 x x x x xt x x t
(18)
where f ( x, t ) f1 ( x x1 ) H1 (t ) f 2 ( x x2 ) H 2 (t ) . Using Eqs. (1) - (4), f1 and f 2
f1 m2 g
m1 ge2
m2u2
m1u1e2
I1
f 2 m3 g
m1 ge1
m3u3
m1u1e1
I1
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can be rewritten as (19)
(20)
2.3. Discretization equations of the load-structure interaction
Using the separation of variables approach, the nonlinear responses of the beam are given by the expression [22, 25]: n
w( x, t ) i ( x)qi (t )
(21)
M
i 1
ED
where qi (t ) are time-dependent generalized coordinates to be determined, i (x) are the linear fundamental vibration modes, and n is the number of vibration modes.
PT
For the present case, the normal modal functions are
2 i sin x , i 1,2,3n AL L
(22)
CE
i ( x)
The vertical displacements, w1 and w2 [see Eqs (5) and (6)], at the contact points
AC
of the wheels and the beam are not independent coordinates, which can be given below
w1 w( x, t ) x x , 1
w1 V
w2 w( x, t ) x x , w2 V 2
w( x, t ) w( x, t ) , x x x t x x 1 1
w( x, t ) w( x, t ) x x x t x x 2 2
(23)
(24)
Substituting Eqs.(19)–(22) into Eq. (18) , Eqs. (5),(6),(23),(24) into Eqs. (1)–(4),
8
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and using Galerkin procedure, the coupled nonlinear ordinary differential equations are obtained in terms of generalized coordinates qi (t ) ( i 1,2,3n ), u1 , , u2 and u3 :
qi (t ) 2ii qi (t ) i2 qi (t )
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[e2i ( x1 ) H1 (t ) e1i ( x2 ) H 2 (t ) ] m1u1 i ( x1 ) H1 (t )m2u2
i ( x2 ) H 2 (t )m3u3 [i ( x1 ) H1 (t ) i ( x2 ) H 2 (t ) ]I1 n
n
n
n
n
n
nol nol k ijkl q j (t )q k (t )ql (t ) cijkl q j (t )q k (t )q l (t ) j 1 k 1 l 1
j 1 k 1 l 1
2 i sin (Vt ) AL L
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(m2 g m1 ge2 ) H1 (t )
(m3 g m1 ge1 ) H 2 (t )
2 i sin Vt AL L
(25)
M
i 1,2,3n
m1u1 (c1 c3 )u1 (c1e1 c3e2 ) c1u2 c3u3
ED
(k1 k3 )u1 (k1e1 k3e2 ) k1u2 k3u3 m1g
(26)
PT
I1 (c1e1 c3e2 )u1 (c1e12 c3e22 ) c1e1u2 c3e2u3
CE
(k1e1 k3e2 )u1 (k1e12 k3e22 ) k1e1u2 k3e2u3 0
(27)
n
AC
m2u2 c2 i ( x1 )qi (t ) c1u1 c1e1 (c1 c2 )u2 k1u1 k1e1 i 1
n k2i ( x1 ) c2V i( x1 )qi (t ) (k1 k2 )u2 m2 g k2 r1 c2r1 i 1
(28)
n
m3u3 c4 i ( x2 )qi (t ) c3u1 c3e2 (c3 c4 )u3 k3u1 k3e2 i 1
n k4i ( x2 ) c4V i( x2 )qi (t ) (k3 k4 )u3 m3 g k4 r2 c4r2 i 1
(29)
where
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i i L
2 EI P L , 1 A EI i
2
i EI i , 2i L A 4
and the nonlinear term coefficients are given by
nol cijkl
3E L i ( x) j ( x) k ( x) l( x)dx 2 0 2 E
L
0
i ( x) j ( x)k( x)l( x)dx
E L ( x) j ( x)k ( x)l( x)dx 0 i
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nol k ijkl
Making the appropriate substitutions and performing the integrations, the coupled nonlinear equations of motion can be written as
MX + (CL + CNol )X + (K L + K Nol )X = F
(30)
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L where M , C and K L are the linear time-dependent mass, damping and stiffness
matrices (i. e., time varying model), respectively, X is the vector of generalized coordinates, given by
qn u1 u2 u3
T
(31)
M
X q1 q2
CNol and K Nol contain the nonlinear terms, which are the nonlinear damping and
ED
nonlinear stiffness matrix, respectively, and F is the generalised force matrix. Eq. (30) consists of n 4 second-order nonlinear ordinary differential equations,
PT
which are solved in the time domain by the Newmark method and the Newton–Raphson iteration method [21]. For the linear time varying model, the
CE
equations of motion can also be obtained from Eq. (30) by setting the nonlinear terms
AC
to zero.
2.4. Surface roughness Surface roughness is one of the important factors of dynamic excitation. The
surface roughness can be represented with a random process that can be described by a power spectral density function. In the present study, the power spectral density function for road surface roughness is given as [26, 27]:
S () S (0 )(
2 ) 0
(32)
10
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where the spatial frequency (cycle/m), S () the power spectral density, 0 -
the reference spatial frequency ( 0 0.1 m 1), S (0 ) is related to the pavement quality, its value is prescribed in Table 1. Surface profiles, r ( x) , are generated as the sum of a series of harmonics: N
r ( x) ak cos(2k x k )
where ak 2S (k ) ,
(33)
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k 1
max min , K min k , min the N -
minimum spatial frequency ( min 0.01m 1), max the maximum spatial frequency ( max 10 m 1), k is the random phase angle uniformly distributed in the interval
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-
[0, 2 ] .
According to Eq. (33), the surface roughness of the beam at the contact points of
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the rear tyre and the front tyre can be obtained as:
r1 (t ) r ( x) x x , r1 (t ) V
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1
r2 (t ) r ( x) x x , r2 (t ) V
(34)
dr ( x) dx x x 2
(35)
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2
dr ( x) dx x x 1
Using Eqs. (34) and (35), the generalized forces caused by the surface roughness can
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be determined in Eqs. (28) and (29). 3. Numerical simulations and results
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3.1 Simple validation of the present method
Example 1 In this section, we present some results about a mass traversing a proportionally damped beam (see Fig.3). The beam is subject to two forces: the weight and the inertia force of moving mass m . From Eqs.(18)-(20), the linear governing equation of the moving mass model is easy to obtain by neglecting the nonlinear terms
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2w w EI 4 w mg m d 2 w( x, t ) C [ ] ( x Vt ) H 2 (t ) d t 2 t A x 4 A A dt 2 where C d
(36)
D , D is the damping coefficient. The linear time varying model can A
be solved by using Galerkin procedure and the Newmark method. This model is similar to the linear model developed by Pesterev and Bergman [28]. The numerical
EI 275.4408m 4 / s 2 , A
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values of the system parameters were as follows: L 6 m,
m 0.2 . Three variants of damping level were considered: (a) C d 0. , (b) AL
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C d 2. , and (c) C d 9. The modes number n used in all of the computations is 3. The displacements curves ( w( , t ) , Vt ) corresponding to these cases are depicted in Fig.4. The comparison shows excellent agreement with Pesterev and Bergman [28].
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Example 2 The moving oscillator model shown in Fig.5 is considered in this section. Using Eqs. (1)–(4), the equations of motion of the oscillator model are derived as
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follows:
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m1z1 c( z1 z 2 ) k ( z1 z 2 ) m1 g f ( x, t )
(38)
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m2 z2 c( z1 z 2 ) k ( z1 z 2 ) m2 g
(37)
where m1 and m 2 are non-suspended and suspended mass of the oscillator model,
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respectively, c the damping coefficient related to suspension, k the stiffness coefficient related to suspension, f ( x, t ) ( x Vt ) the interaction force at the wheel, and z1 and z 2 is the vertical displacement of the non-suspended mass m1 and suspended mass m 2 , respectively. From Eq.(18), the linear governing equation of the beam is easy to obtain by neglecting the nonlinear terms
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A
2w w 4w C EI f ( x, t ) ( x Vt ) H 2 (t ) d t 2 t x 4
(39)
Using Eqs. (37) and (38), the interaction force is given as
f ( x, t ) (m1 m2 ) g (m1 z1 m2 z2 )
(40)
where the vertical displacement, z1 , at the contact point of the oscillator model and
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the beam is not independent coordinate, which can be given below
z1 w( x, t ) x Vt w( x, t ) x
z1 V 2
x Vt
2 w( x, t ) x 2
w( x, t ) t
2V
x Vt
(42)
x Vt
2 w( x, t ) xt
2 w( x, t ) t 2
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z1 V
(41)
x Vt
(43)
x Vt
Numerical results, based on the coupled linear time varying mode (see Eq.(30), setting the nonlinear terms to zero), have been computed and plotted in order to
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establish reasonable comparisons. The following dimensionless parameters [11] are used:
(m1 m2 ) ) , 0.3 ( b1 , v v AL
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(
EI V , b1 2 ) , b1 L AL4
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0.5 (
0 0.25 ( 0
m1 ) , 0.5 m2
k1 c ) , v 0.125 ( v ) , m2 2m 2 v
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and the modal damping ratio is set to be 0.02. The response magnification factor is
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used in the results, which is defined as
Dd
w( x, t ) Dst
( Dst
(m1 m2 ) gL3 ) 48EI
(44)
As can be seen from the comparisons (see Fig.6), the agreement with those in the literature [11] is good. 3.2 Nonlinear vibration of the load-structure interaction Several examples of applications of the methodology described in the above are presented here. A sensitivity study was conducted to analyze the effect of key
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parameters on the nonlinear vibration of the system. These parameters included the roughness coefficient, the span, prestress, internal damping of the beam and moving speed of the load. Table 2 was the mechanical properties used in all of the computations unless otherwise specified. Pcr
2 EI L2
is Euler’s buckling load. In all
of numerical examples, zero initial conditions were assumed.
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The linear and nonlinear dynamic responses computed for the beam and vehicle have been plotted in Figs. 7 and 8. Fig. 7 shows the effect of the nonlinear on the displacements, velocities and accelerations of the beam at the midspan. Fig. 8 shows the effect of the nonlinear on the vertical displacements, vertical velocities and
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vertical accelerations of the vehicle body. It can be seen that the dynamic responses of the nonlinear analysis is higher than the one obtained from the linear solution. This phenomenon was also noted elsewhere [22]. This incident is known as the softening behavior which is mostly due to the existence of the cubic non-linearities in the
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equations of motion of the load-structure system, and hence this system is taken to be equivalent to a nonlinear soft spring [25]. A free vibration with damping (decay) can
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also be observed after the load departs from the beam in Fig. 7. Form this, one may also conclude that the nonlinear analyses are more expensive than the linear approach
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with respect to both storage and CPU time. For different surface roughness of the beam, Fig. 9 displays the nonlinear
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acceleration responses of the beam at the midspan and the vertical acceleration of the vehicle body. It is observed that the nonlinear acceleration responses of the beam and
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the vertical acceleration of the vehicle body increase as the surface roughness increases (road class:B→C→D, see Table 1). This is due to the fact that the dynamic excitation increases as the surface roughness coefficient increases. The impact factor is an important parameter in the design of beams, and can be defined [30] as follows:
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Rd R s Rs
(45)
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where Rd and Rs are the absolute maximum responses at the midspan from the dynamic and static studies respectively. Fig.10 shows that the impact factor at the midspan decreases as the span L of the beam increases. The impact factor decreases rapidly as the span L increases from 0 to 14m. However, when L is larger than 16m, the impact factor changes very slowly, and the impact factors are almost same for
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different internal damping of the beam. From this figure, it can also be said that by increasing the value of internal damping , the impact factor at the midspan decreases which is generally a natural phenomenon in any structural system. Designers may obtain the desirable dynamic characteristics adequate to design
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purpose as they choose the span L and internal damping appropriately.
Fig.11 illustrates the linear and nonlinear dynamic responses at the midspan of the beam for different prestress loads. It is clear to see that the impact factor at the midspan increases with the increase of the prestress load. This phenomenon was also noted
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elsewhere [31]. When P / Pcr >0.35, the difference between the linear and the
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nonlinear model increases with the increase of the prestress loads. This is due to the fact that the stiffness of the beam will be reduced due to the influence of prestress, so
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the beam is softened and the nonlinearity of the system increases. The impact factor is sensitive to the moving speed of the load. Fig.12 shows the
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effect of the moving speed of the load on the impact factor. In this figure, the load velocity ranges from 5m/s to 85m/s. The maximum impact factor shows a peak,
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which correspond to the critical velocity Vcr 50 m/s. It can be said that in the under critical velocity ( V < Vcr ) the impact factor of the beam generally increases by increasing the moving speed of the load, and in the overcritical velocity ( V > Vcr ), the impact factor decreases by increasing the moving speed of the load. The phenomenon is also reported in linear moving force model [7]. According to the governing equations (25), the critical velocity can also be defined as
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Vcr
L ( i 1,2,3 ) i
(46)
where is the natural frequency of the load-structure system (see Eqs.(25)-(29), neglecting the nonlinear terms ). The result calculated from Eq. (46) is agreement with the numerical result in Fig.12. 4. Conclusions
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This paper focuses on the nonlinear vibration and dynamic interaction of the beam under moving loads. The numerical results show that the dynamic responses of the nonlinear analysis are higher than the one obtained from the linear solution, and the nonlinear responses of the beam and the vehicle body increase as the surface
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roughness increases. The results also confirm that the impact factors decrease rapidly as the span and the damping increase. However, when the span is larger, the impact factor changes very slowly, and the impact factors are almost same for larger damping. Another finding is that the nonlinear dynamic responses, and the difference between the the linear and the nonlinear model increase with the increasing of the prestress
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load. Moreover, the critical velocity can be observed by the simulation results. The
of beams.
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Acknowledgements
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meaningful results in the present paper are helpful for the application and the design
The authors thank the supports of the Hunan Provincial Natural Science Foundation
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of China under No. 13JJ4053.
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References
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variable thickness circular arches traversed by a moving point load, Appl. Math. Model. 40 (2016) 4640–4663. [4] S.S. Law, X.Q. Zhu, Bridge dynamic responses due to road surface roughness and braking of vehicle, J. Sound Vib. 282 (2005) 805–830. [5] K. Keivan, G.A. Hamidreza, N.K. Ardeshir, On the role of shear deformation in
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[9] C. Johansson, C. Pacoste, R. Karoumi, Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by
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concentrated moving loads, Comput. Struct. 119 (2013) 85–94. [10] H. Ding, K.l. Shi, L.Q. Chen, S.P. Yang, Adomian polynomials for
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[11] D.Y. Zhu, Y.H. Zhang, H.A. Ouyang, Linear complementarity method for dynamic analysis of bridges under moving vehicles considering separation and surface roughness, Comput. Struct. 154 (2015) 135 –144.
[12] Z. Dimitrovová, Critical velocity of a uniformly moving load on a beam supported by a finite depth foundation, J. Sound. Vib. 366 (2016) 325–342. [13] X.Q. He, M. Rafiee, S. Mareishi, K.M. Liew, Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams, Compos. Struct. 131 (2015) 1111–1123. 17
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[14] M. Şimşek,
Nonlinear static and free vibration analysis of microbeams based
on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Compos. Struct. 112 (2014) 264–272. [15] Kefei Zhu, Jintai Chung, Nonlinear lateral vibrations of a deploying Euler–Bernoulli beam with a spinning motion, Int. J. Mech. Sci. 90 (2015) 200–212.
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[16] Jianshe Peng, Liu Yang, Fan Lin, Jie Yang, Dynamic analysis of size-dependent micro-beams with nonlinear elasticity under electrical actuation, Appl. Math. Model. 43 (2017) 441–453.
[17] Da Chen, Sritawat Kitipornchai, Jie Yang, Nonlinear free vibration of shear
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deformable sandwich beam with a functionally graded porous core, Thin-Walled Struct. 107 (2016) 39–48.
[18] M. Faraji Oskouie, R. Ansari, Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects, Appl.
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Math. Model. 43 (2017) 337–350.
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slender meso-periodic beams.The tolerance modeling approach, Compos. Struct. 136 (2016) 270–277.
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[20] S.S. Law, X.Q. Zhu, Nonlinear Characteristics of Damaged Concrete Structures under Vehicular Load, J. Struct. Eng. 131(8) (2005) 1277–1285.
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[21] M. Şimşek, T. Kocatürk, Nonlinear dynamic analysis of an eccentrically prestressed damped beam under a concentrated moving harmonic load, J. Sound
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Vib. 320 (2009) 235–253.
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[23] C. Tao, Y.M. Fu, H.L. Dai, Nonlinear dynamic analysis of fiber metal laminated beams subjected to moving loads in thermal environment, Compos. Struct. 140 (2016) 410 – 416. [24] A. H. Karimi, S. Ziaei-Rad, Vibration analysis of a beam with moving support subjected to a moving mass traveling with constant and variable speed, Commun. 18
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Nonlinear Sci. Numer. Simulat. 29 (2015) 372–390. [25] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, 1979. [26] O. Javier, M.G. José, A. Pablo, Á.A. Miguel, Relevance of a complete road surface description in vehicle-bridge interaction dynamics, Eng. Struct. 56 (2013) 466–476.
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[27] ISO-8608. Mechanical vibration-road surface profiles-reporting of measured data.
[28] A.V. Pesterev, L.A. Bergman, Response of a nonconservative continuous system to a moving co ncentrated load, J. Appl. Mech. 65 (1998) 436 – 444.
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[29] S.S. Law, J.Q. Bu, X.Q. Zhu, S.L. Chan, Vehicle axle loads identification using finite element method, Eng. Struct. 26 (2004) 1143–1153.
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[31] H. Zhong, M.J. Yang, Z.L. Gao (Jerry), Dynamic responses of prestressed bridge and vehicle through bridge-vehicle interaction analysis, Eng. Struct. 87 (2015)
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116–125.
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Fig. 1. Model of the load-structure system.
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Fig. 2 Differential element of the beam of length dx .
Fig. 3. Model of the moving mass system.
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0.30
(a )
w( , t) (m)
0.25
Present work Pesterev et al.( 1998)
0.20 0.15 0.10
0.25
(b )
0.2
w( , t) (m)
0.15 0.10
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0.05 0.2
0.4
w( , t) (m)
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0.12
0.6
0.8
1.0
0.6
0.8
1.0
0.8
1.0
t
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(c )
t
Present work Pesterev et al.( 1998)
0.20
0.00 0.0
0.4
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0.00 0.0
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0.05
Present work Pesterev et al.( 1998)
0.09 0.06 0.03 0.00 0.0
0.2
0.4
0.6 t
Fig. 4 Displacements of the mass moving with constant speed v 6 m/s on the simply supported beam: (a) C d 0.
(b) C d 2.
(c) C d 9. 21
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1.8 1.5
Present work Zhu et al. (2015)
1.2 0.9 0.6 0.3
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0.0 0.0
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Response factor Dd
(a )
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Fig.5. Model of the moving oscillator system.
0.2
0.4
0.6
0.8
1.0
0.8
1.0
t
2.5
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Response factor Dd
(b )
2.0
Present work Zhu et al. (2015)
1.5 1.0 0.5 0.0 0.0
0.2
0.4
0.6 t
Fig. 6. Response magnification factors of system: (a) displacement at the contact point of the wheel and the beam, (b) displacement at the midspan of the beam.
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0.04
(a )
-0.04 linear nonlinear
-0.08 -0.12 0.0
0.5
1.0
1.5 t (s)
2.5
3.0
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(b )
2.0
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w( L/2,t) (m)
0.00
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(c)
Fig. 7. Time history responses of the beam midspan: (a) vertical displacement, (b) vertical velocity, (c) vertical acceleration.
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(a )
linear nonlinear
0.00
u1(m)
-0.05 -0.10
-0.20 0.0
0.3
0.6
0.9 t (s)
1.2
1.5
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(b )
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-0.15
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(c )
Fig. 8. Time history responses of the vehicle body: (a) vertical displacement, (b) vertical velocity, (c) vertical acceleration.
24
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(a )
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(b)
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Fig. 9 Effect of the surface roughness of the beam on the time history responses: (a) vertical acceleration of the vehicle body, (b) vertical acceleration of the beam midspan.
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= 0.05 = 0.07 = 0.09
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0.9
0.6
0.3 12
15 L (m)
18
21
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9
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1.2
Fig.10. Effects of the span L and internal damping of the beam on impact
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factors.
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0.6
0.4
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0.5
linear nonlinear
0.3 0.2 0.1
0.2
0.3
0.4
P/Pcr
Fig.11. Effect of the prestress load of the beam on impact factors.
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1.0 linear
nonlinear
0.8
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0.6 0.4 0.2
10 20 30 40 50 60 70 80 90 V (m/s)
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0
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Fig.12. Effect of the moving speed of the load on impact factors.
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Table 1. Classification of road roughness [26]. S (0 ) (m3)
road class
16×10
B (Good)
64×10
-6 -6
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A (Very good)
C (Medium)
256×10
-6
1024×10
E (Very poor)
4096×10
-6 -6
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D (Poor)
Table 2. Parameters of the load-structure system [21, 29]. Beam
Vehicle
I1 1.47 105 kgm2
m1 17735 kg
4.27 m
e1 2.05 m
e2 2.22 m
m2 1000 kg
0.02 s
m3 1500 kg
k1 2.47 106 N/m
P 0.2 Pcr
k2 3.74 106 N/m
k3 4.23 106 N/m
A 0.36 m2
k4 4.60 106 N/m
c1 4.00 104 N/m/s
I 0.0192 m4
c2 4.30 103 N/m/s
c3 3.00 104 N/m/s
S (0 ) 64 106 m3
c4 3.90 103 N/m/s
V 15 m/s
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A 1500 kg/m
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E 35 Gpa
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L 20 m
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